Quasicrystalline 30 Twisted Bilayer Graphene As an Incommensurate

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Downloaded by guest on October 1, 2021 uscytl ihacascddcgnlpten(12). pattern dodecagonal classic rota- a the 30 with quasicrystals preserving example, for while symmetry; space tional real long-ranged in to a symmetry lacks contrast superlattice translational incommensurate In 1B), Fig. exam- (11). in (see ple period common long-ranged a with more superlattice commensurate is superlattice mensurate tBLG commensurate forming 0 for angles for twisting possible the where www.pnas.org/cgi/doi/10.1073/pnas.1720865115 Avila Jose Yao Wei coupling interlayer strong with superlattice incommensurate an 30 Quasicrystalline r,rsetvl,and respectively, ers, graphene bilayer twisted to engineering quasicrystalline structure superstructures. incommensurate band incommensurate in extending important thereby the coupling highlights superlattices, interlayer vector work lattice Our of reciprocal layer. role the graphene by other layer gen- the a graphene of a one of via in scattering boundary coupled cone is, zone Dirac are mechanism—that the layers scattering at graphene Umklapp opening of eralized two gap zone these a Brillouin that and the suggest layer inside cones graphene Dirac each mirrored cm interval- of 1383 the at emergence and mode diffraction Raman electron double-resonance energy ley low 30 30 by The confirmed structure. quasicrystalline electronic is coherence. its of reveal phase and growth substrate, of (30 Pt(111) successful lack graphene bilayer the the twisted to report been due we often negligible has Here interaction be interlayer to the only also assumed not but not are nature, quasicrystals) in but to rare analogy symmetry (in Moir rotational symmetry translational commen- long-ranged with on So heterostructures a material. focused mensurate with single been superlattices a has to in heterostructures surate (superlattices) possible in not research electronic heterostructures are far the Waals that engineer der 2017) properties to 1, obtain December van used review for be of (received 2018 can 25, structure May coupling approved and interlayer CA, Stanford, The University, Stanford Chemistry, of Department Dai, Hongjie by Edited France; Cedex, Yvette and sur Gif 91192 48, Aubin-BP China; 100084, Beijing a iae rpee(BG a evee sasml eso of version simple a as viewed with be “hetero”-structure, can (tBLG) graphene bilayer a heterostructure aifis(,10) (9, satisfies to respect m I tt e aoaoyo o iesoa unu hsc,Tigu nvriy ejn 004 China; 100084, Beijing University, Tsinghua Physics, Quantum Dimensional Low of Laboratory Key State ◦ 1,2 bottom a e al eeotutrs() ogrne commen- long-ranged a Moir (1), heterostructures surate Waals der van n a hl o agrtitage,epcal rud30 around especially angles, twist larger for while , 1 top f q olbrtv noainCne fQatmMte,Biig108,P .China R. 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Direct PNAS a is article This interest. of conflict no declare authors and The J.Z., W.Y., and data; paper. analyzed the S.Z. wrote and C.C., W.Y. S.Z. C.K.C., research; K.B., performed K.Z., J.Z. Y.Z., and C.B., M.C.A., E.W., W.Y., J.A., research; designed S.Z. contributions: Author owo orsodnemyb drse.Eal yhumi.snhaeuc or [email protected] Email: addressed. be [email protected] may correspondence whom To a,b,f,1 plcto ag fvndrWashtrsrcuefrfuture for heterostructure devices. Waals electronic structure, der the van broaden incommensurate of results range These application the before. overlooked interaction, in been interlayer has which importance the of its consequence showing a are mirrored these cones that “het- identify Dirac We cones. incommensurate Dirac emer- mirrored the for of report gence we example order, quasicrystalline an with erostructure” as ques- (tBLG) 30 fundamental growing successfully graphene a by of work, is this structure heterostructure In tion. electronic nonperiodic the a affects such interaction of How assumption electrons. interlayer the of the and movement nature interlayer in coherent rarity suppressed commensu- the heterostructures to been Unlike due often Waals neglected, have phenomena. ones incommensurate der exotic heterostructures, rate many van induce in could interaction Interlayer Significance lhuhteeetoi tutrso omnuaehet- commensurate of structures electronic the Although a,b ◦ tL ucsflyo t11 usrt n reveal- and substrate Pt(111) on successfully -tBLG ei Bao Kejie , NSlicense. PNAS b d eateto hsc,Tigu University, Tsinghua Physics, of Department ycrto OEL ’redsMrses Saint Merisiers, des L’Orme SOLEIL, Synchrotron c hnKiChan Kai Chun , . www.pnas.org/lookup/suppl/doi:10. NSLts Articles Latest PNAS c houChen Chaoyu , ◦ titdbilayer -twisted ◦ | -tBLG d f6 of 1 ,

APPLIED PHYSICAL SCIENCES A incommensurate C incommensurate heterostructure work shows that distinguished from other graphene/Pt with com- commensurate (Quasicrystalline) mensurate (e.g., 2 × 2, 3 × 3) Moiré superstructures (21), the ◦ 0 60 interface between the 30 monolayer graphene and the Pt(111) substrate is incommensurate without forming any Moiré pattern, leading to nearly free-standing monolayer graphene (22). Fur- ther increasing the annealing temperature and time can lead to a thicker graphene sample with a new set of diffraction peaks emerging at 0◦ orientation (red arrow in Fig. 2B) coexisting with those at 30◦, and such a graphene sample is the focus of the cur- B rent work. The lattice constants extracted from LEED show that commensurate superlattice there is negligible strain (<0.2%; see SI Appendix, Fig. S1 for details) between the 30◦ and 0◦ graphene layers, and the absence of additional reconstructed diffraction spots in the LEED pattern further supports that they do not form commensurate superlattice. If these diffraction peaks come from the same graphene domains, this would imply that 0◦ graphene is stacked on top of the bottom 30◦ graphene layer—namely, a 30◦-tBLG is formed. In the following, we will provide direct experimental evidence Fig. 1. Commensurate and incommensurate tBLG. (A) The distribution of for the conjectured incommensurate 30◦-tBLG from Raman all possible twisting angles θt for forming commensurate tBLG when q = 1 spectroscopy and reveal its electronic structure from ARPES in Eq. 1. The Moire´ period at corresponding twisting angles is represented measurements. by the length of the solid line (with logarithmic scale). (B) Commensurate tBLG with 7.34◦ twisting angle (indicated by the blue arrow in A), showing Moire´ pattern with periodic translational symmetry. Green arrows are Intervalley Double-Resonance (DR) Raman Mode. Raman spec- the Moire´ reciprocal lattice vectors. (C) Incommensurate 30◦-tBLG without troscopy is a powerful tool for characterizing the vibrational any long-ranged period (indicated by red arrow in A), showing patterns of mode in graphene (23) and can provide direct information about dodecagonal quasicrystal. the sample thickness and stacking. Fig. 2G shows a typical optical image of the as-grown sample. The optical image shows strong intensity contrast, with a darker region of ≈10 µm overlapping heterostructure is stabilized under appropriate substrate with the brighter region. The Raman spectra in Fig. 2C show conditions. stronger intensity for the darker region (red curve) than the brighter region (blue curve), suggesting that the darker region Results is thicker. The spectra for both regions show characteristic fea- Growth of 30◦-tBLG. The 30◦-tBLG sample was grown on the tures of monolayer graphene (23), in which the 2D mode shows a Pt(111) substrate by carbon segregation from the bulk sub- single Lorentzian peak with stronger intensity than the G mode. strate (21, 22). Fig. 2A shows the low energy electron diffraction This suggests that the top and bottom flakes are both monolayer (LEED) of bottom monolayer graphene at 30◦ azimuthal ori- graphene, and the thicker region is not a Bernal (AB stack- entation (blue arrow) relative to the substrate. Our previous ing) bilayer graphene but is rather a bilayer graphene with a

A 1200 °C CD E F Monolayer 2D Ktop Mbottom Mbottom K'top DR G Bilayer bottom q=Gbottom hν 30° E Pt q -Qphonon -Q G phonon Intensity (a.u.) Intensity (a.u.)

hνS 1000 1500 2000 2500 3000 1300 1350 1400 -1 -1 Raman Shift (cm ) Raman Shift (cm ) 1600 °C BG H I J 1600 )

-1 TO 3 μm High 1500 0° 30° Q 1400 phonon -1 1383 cm 1300 Low phonon frequency (cm 1200 -1 -1 D peak (1353 cm ) R peak (1383 cm ) Γ Κ Μ Κ' Γ

Fig. 2. Observation of DR Raman mode in 30◦-tBLG. (A) The LEED pattern for the graphene sample after annealing at 1,200 ◦C measured at electron beam energy of 135 eV. (B) The LEED pattern for the graphene sample at the higher annealing temperature of 1,600 ◦C. (C) Measured Raman spectrum for monolayer and bilayer region, respectively, covering the range of G peak and the 2D peak. (D) Zoom in of C for a range from 1,300 to 1,430 cm−1.(E) The geometry for electron momentum transferring in the DR Raman process of the observed R mode. (F) The schematic drawing for the DR process of R mode in graphene band structure (not in scale). An electron at one Dirac cone from the top layer is photo-excited to the conduction band, scattered by one reciprocal lattice vector of the bottom layer q = Gbottom, and subsequently scattered back by phonon with vector Qphonon.(G) The optical image for the measured area. The red and blue dots indicate the measuring positions of spectra in C and D. (H) The Raman map for the D peak, integrating intensity from 1,343 to 1,363 cm−1. (Scale bar: 3 µm.) (I) The Raman map for the R peak, integrating intensity from 1,373 to 1,393 cm−1. (Scale bar: 3 µm.) (J) The phonon spectrum of the TO mode along high symmetric direction, taken from ref. 26. The arrow indicates the momentum of transferring R mode phonon. Inset shows the high-symmetry path in k-space for the phonon spectrum. The left hexagon is the ﬁrst Brillouin zone (BZ).

2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1720865115 Yao et al. Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 a tal. et Yao nrysmlrt h ia oe at binding cones high Dirac to the low to from similar expand visi- The contours energy higher these scale. with that energies logarithmic show different bility the at 3C) using (Fig. images features curvature weak near the BZ enhancing detected first are the intensity weaker inside with contours Additional pattern. 0 of istence h ioemti lmn 2) ihsrne nest between intensity by stronger with caused (27), asymmetry K element intensity matrix similar dipole at shows the cone cone Dirac Dirac the rored of image reflected the near persion as of position reflected as (labeled layers vec- with lattice layer reciprocal graphene one bottom 2 by the connected of opposite are two tor between that scattering by cones caused the is Dirac that case claim our we in instead mode R cones, vector Dirac wave neighboring two with between phonon a emitting q nonzero a by by valley valley inal another to scattered transfer are momentum valley one (by interval- at process Moir trons the scattering or special for a phonon, process involves defect, key process intervalley The Raman mode: 26). DR D the ley (25, as process origin similar Raman a DR has likely mode R the defect or contrast, tBLG. region In the 2I peak. (Fig. bilayer cm region D graphene the 1,383 the bilayer at defects of of peak nature or edges the the 2H the size with (Fig. at consistent limited peak centers, appears the this only for by mapping it caused Raman is The cm (23). 1,353 which at graphene, peak of The 2D). cm 1,353 (Fig. at respectively centered peaks two shows region n r o bevdi h otmmnlyr30 monolayer bottom open- the gap and in cones observed Dirac not mirrored are Such black by 3D). ing (pointed Fig. opening in gap a arrows indicates cones point Dirac crossing mirrored the at (more the intensity mode in at Raman trans- shown DR the cone are the to details for Dirac related above original discussed strongly mechanism is the fer and from valley momentum comes opposite it that suggesting 0 both for (BZ) zone Brillouin graphene the of a points with ARPES at conventional map of by size measured first beam was Cone. sample Dirac the Mirrored of Observation 30 the and next. of investigate below structure will electronic discussed we exotic which be the to by closely interaction supported is is interlayer mode R the 30 observed to the interesting related for this process in scattering layer novel This bottom lattice the reciprocal of the involving vector process 30 scattering incommensurate unreported an an the Moir is that a suggesting region not superlattice, bilayer commensurate is any mode of R vector the cal of momentum phonon (TO) The optical the transverse along the spectrum in phonon energy and momentum matching cm 1,430 and 1,300 between spectra zoom- the when in importantly, More ing (24). instead angle twisting large pera h Zbudr ftebto ae n saway is and layer bottom the of (see boundary bands substrate BZ the from the at appear (see 0 seFg 2F Fig. (see E ic h nryo h oei ls ota fteDmode, D the of that to close is mode R the of energy the Since ◦ K and R IAppendix SI (K ihrsett h 30 the to respect with ) . V hrceitccnclcnor pera h K the at appear contours conical Characteristic eV. −0.8 R F n h on fte30 the of point M the and ) .Ti svrfidb bevn hnnmd with mode phonon a observing by verified is This ). ◦ n 30 and - .WieDmd sidcdb eetscattering defect by induced is mode D While ). ∼100 o eal) utemr,tegphpesto happens gap the Furthermore, details). for K K R K 0 q .Fg 3A Fig. µm. .Mroe,tesprsinof suppression the Moreover, Discussion). ◦ ◦ −1 0 lososlna eair hc sthe is which behavior, linear shows also n usqetydcybc oteorig- the to back decay subsequently and gahn aesosre nteLEED the in observed layers -graphene ◦ and ´ atr)i hc htectdelec- photoexcited which in pattern) e idctdb re ro n labeled and arrow green a by (indicated “”pa)i bevdi h whole the in observed is peak) (“R” K 0 K Γ-K-M-K ◦ IAppendix SI 30 ◦ when 3B) Fig. in arrows (green ◦ M ,idctn hti sitiscto intrinsic is it that indicating ), Zegs(lelns.Tedis- The lines). (blue edges BZ ,i gemn ihtecoex- the with agreement in ), 30 hw h RE intensity ARPES the shows ◦ h lcrncsrcueof structure electronic The K 0 ewe h rgnland original the between -Γ K 0 ◦ ◦ ◦ 0 gahn layer -graphene q n hyocra the at occur they and , ieto seFg 2J Fig. (see direction tL.I loreveals also It -tBLG. ◦ o eal) suggesting details), for = .Temir- The 3D). (Fig. −1 −1 G n ,8 cm 1,383 and bottom ◦ −1 steDmode D the is tL region. -tBLG hw that shows ) h bilayer the , ◦ Q -graphene ´ recipro- e ◦ phonon seFig. (see ◦ n 30 and -tBLG, M 30 −1 = ◦ ). ◦ , , ia oe nedoiiaefo hsnwydsoee tBLG discovered newly this from originate indeed 30 cones the Dirac for evidence defini- experimental provide tive resolution spatial with NanoARPES by measured the along 30 that bottom the with 30 coexisting the the layer from along dispersions graphene dispersion shows C with region tBLG, while 4B), (Fig. layer the dis- shows in out along B map by shown persion labeled and Region is regions structure. different the map electronic distinguish distinct intensity their can at we resolved where resolution of spatially 4A, structure A Fig. spatial scale. band with nm electronic ≈120 structures the resolving graphene of coexisting capable is which 30 of Resolution. Spatial ture with Structures Band The 30 other. 30 0 the each the the interac- that with of overlaps graphene–substrate confirm tially properties results the ARPES intrinsic by our by not but caused tion are they that cone. Dirac replica emerging the and original the between (E relation arrows. mirrored 0 horizontal the the by 0 of indicated of 30 is point point of K K point the M the indicate the arrows near vertical dispersion three Dirac-type The orientation. measured The (D) lines. E 0 from of energies boundary at BZ rectangle the by marked (C region cones. Dirac (Left replica BZ of the traces the indicate arrows (B lines. dashed at 3. Fig. DE C AB E-E (eV) -1 . V h Z f0 of BZs The eV. −0.8

-1.5 F-1.0 -0.5 -2.0

0.0 k (Å ) 2.0 0.5 1.0 y 1.5 -0.8 eV bevto fmroe ia oei 30 in cone Dirac mirrored of Observation 1.2 -1.0 n h Dcraueiaeo osateeg asfrthe for maps energy constant of image curvature 2D the and ) ◦ K K tL sfrhrcnre yNnAPS(28), NanoARPES by conﬁrmed further is -tBLG R 30° h aempas map same The ) ◦ ◦ 1.4 k k -0.5 x 30 , BZ(replica) BZ(0°) BZ(30°) ◦ .Teeoe dispersions Therefore, 4C). (Fig. layer graphene (Å // (Å gahn,adteKpito h elc Z h gap The BZ. replica the of point K the and -graphene, M -1 ◦ 30° -1 ) Mdrcino h 30 the of direction Γ-M n elc ad r hw srd le n green and blue, red, as shown are bands replica and , ) 1.6 ◦ gahn n 30 and -graphene 0.0 K K 0° ◦ 0° 1.8 rpeelyradte neatwith interact they and layer graphene High Low A u ihtelgrtmcsae Green scale. logarithmic the with but ceai lutainsoigthe showing illustration Schematic )

Kdrcino h 0 the of direction Γ-K -1

0.5 1.0 k (Å ) 1.5 2.0

◦ y gahn r niae ythe by indicated are -graphene ◦ k tL,adtemirrored the and -tBLG, (log scale) -0.8eV x NSLts Articles Latest PNAS ◦ K K ◦ -1.0 R F h lcrncstruc- electronic The R nest map Intensity (A) -tBLG. rpeelyrspa- layer graphene ◦ ceai rwn of drawing Schematic ) ◦ to tL.Therefore, -tBLG. E otmgraphene bottom . V(Right eV −0.8 Mdrcinof direction Γ-M M k 30° -0.5 x M (Å 30° -1 ) K ◦ 0° -graphene, 0.0 K 0° | ◦ .The ). f6 of 3 0 eV top -0.8 -0.4 k ◦ y ◦ -

APPLIED PHYSICAL SCIENCES A B High B

C 2 m Low

k1 k13

BCD0 M30° 0 KR M30° K0° k13 -1 -1 Fig. 4. Conﬁrmation of 30◦-tBLG and the electronic structure by NanoARPES measurements. (A) Spatial intensity map obtained by integrating the energy -2 (eV) -2 (eV) F F from −1.5 eV to EF . The structures for regions B and E-E E-E -3 -3 C are schematically shown on Right. (B and C) Dis- Intensity (a.u.) persion cuts taken at regions marked by B and C in A. (D) The energy distribution curves (EDCs) taken in -4 -4 the momentum range marked by the arrows in C. k 1 The EDC through the cross point of the original and 1.0 1.2 1.4 1.6 1.8 1.0 1.2 1.4 1.6 1.8 -4 -3 -2 -1 0 -1 -1 mirrored bands is shown by the red color. Blue and k// (Å ) k// (Å ) E-EF (eV) green circles mark the peak positions.

structure. Moreover, by focusing only in the 30◦-tBLG region, Band Gap in k Space. To track the evolution of the gap at the sharper dispersions can be obtained, and the gap at the M point crossing points between the original Dirac cone at K0◦ and the ◦ becomes more obvious. Analysis from the EDCs for 30 -tBLG mirrored Dirac cone at KR, we show in Fig. 5A dispersions mea- (Fig. 4D) shows a gap at the M30◦ point, suggesting the Dirac sured from cuts 1 to 5 (labeled in Fig. 5B). The gap size is cone at K0◦ and the mirrored Dirac cone are hybridized. quantiﬁed by the peak separation of the EDCs at the crossing

A 12345

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-2.0 1.4 1.6 1.4 1.6 1.4 1.6 1.4 1.6 1.4 1.6 -1 -1 -1 -1 -1 kx (Å ) kx (Å ) kx (Å ) kx (Å ) kx (Å ) 1.8 -1.60 B C D Experiment E K0° 280 Fitting Curve

large 1.6 260 -1.65 ) -1

(Å 240 x M30° k 1.4 -1.70

Gap Size (meV) 220 1 5 Gap Position (eV) = 3.10 eV 0 K 6 R = 1.00×10 m/s small 1.2 200 F -1.75 -0.2 0.0 0.2 0.00 0.10 0.00 0.10 -1 -1 -1 ky (Å ) ky (Å ) ky (Å )

Fig. 5. Evolution of gap size and gap position in k space. (A) Five different cuts located at different positions in the k space shown in B. The EDCs crossing the gap are shown on the right of the spectrum as red symbols. The fitting curves are plotted as blue lines, and green curves are the two Lorentzian peaks from fitting results. Black markers show the positions of the two peaks. (B) Schematic BZ showing the positions of the five cuts in A. (C) The gap size along the ky direction. (D) The gap position along the ky direction. The green line is the fitting curve by the tight-binding model. (E) Cartoon diagram for the two Dirac cones and their intersecting line with a gap. The color indicates the gap size.

4 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1720865115 Yao et al. Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 a tal. et Yao tutr n nifiieylrecl sbyn h calcula- the beyond is of supercell cell finite large a infinitely capability, tion nonperiodic an the to and apply not structure does theorem 30 Bloch Since of stability. structure electronic the reveal Calculation. Theoretical shown is cones Dirac 5E. mirrored Fig. and in original the between gap the 1.003 be to extracted tutr f30 of structure between state at bonding and clear balls) state a Bloch showing the level, of Fermi projection Real-space of (B) gap indicated. energy The scheme. (5 a of structure Band (A) 6. Fig. occurs coupling p–d such 6B) Fig. Since in level. (shown Fermi between orbital d near coupling platinum sub- revealed significant and Pt is a orbital the p states, carbon of Bloch the role the 30 important of the an projection lower stabilizing suggests atom) in 30 and strate meV/(C why sample 4.1 explains our of which in one, energy AB-stacking sub- formation the than to of inclusion formation leads BLG, AB-stacking the strate common While more the calculation. 30 than the free-standing interest- of in 30 energy An included gap of measurements. is the stability with the substrate experimental layer, regards the counterpart observation the ing to of close point M value 6A the Fig. at in opening 30 structure nonperiodic electronic the calculated mimic to constructed is positions model gap 5 tight-binding the Fig. a fit in from to obtained possible dispersion Fig. (see is energy in it the curve hence, by position cones; aver- gap Dirac the The as two positions. position peak gap two the 5D the define of can we age addition, of In value point. maximum the that the from clear at decreases is It size 5C. Fig. gap in plotted the is value extracted the and points, hc scoe otevledrvdfo hoeia calcula- theoretical from velocity derived Fermi value The the (29). parameter to tions closed hopping is neighbor which nearest a giving AB C E-EF (eV) -8.0 -6.0 -4.0 -2.0 0.0 2.0 hw httegpi oae tteitretn ieo the of line intersecting the at located is gap the that shows IAppendix SI K' M lcrncsrcueo 30 of structure Electronic 0º d 30 si odareetwt h xeietldata, experimental the with agreement good in is D) ria fP wiebls.(C balls). (white Pt of orbital 180 meV ◦ ◦ on to point tL,soigteeegneo irrdDrccones. Dirac mirrored of emergence the showing -tBLG, o eal) h tigcre(re curve (green curve fitting The details). for ±0. × 0 e hndvaigfo the from deviating when meV ∼200 5)/(3 biii acltosaepromdto performed are calculations initio Ab 002 π bn tteMpito h te ae is layer other the of point M the at -band ◦ 30° graphene ◦ 117 meV √ 0° graphene tL s16mV( tm higher atom) meV/(C 1.6 is -tBLG tL.B netgtn real-space investigating By -tBLG. × 3 ◦ tL rmfis-rnilscalculations. first-principles from -tBLG × 10 G ν 3 6 F √ 30º ceai lutainfrteband the for illustration Schematic ) /.Ashmtcsmayof summary schematic A m/s. 3)R30 = ◦ (5 tL n netgt the investigate and -tBLG √ 3a × ◦ tL nteetne zone extended the in -tBLG 5)/(3 γ ◦ hw h adgap band the shows 0 ◦ tL hntePt the when -tBLG /2 tL ytm The system. -tBLG p K z ◦ R ria fC(brown C of orbital √ (a tL emerges -tBLG γ Γ 3 0 ∼ × on erthe near point f31 eV, 3.10 of 8 meV ∼280 3 2.46 √ K 3)R30 0º M )is A) ˚ 30 ◦ ◦ t11 usrt oaot1,600 about to substrate Pt(111) Growth. Sample Methods and Materials Waals der commen- van other both as of well heterostructures. structure as tBLG band incommensurate the and mechanism engineer surate scattering to applied a be Such can strong superlattice. such the in incommensurate process for scattering an coherent evidence 30 a a through experimental coupling in direct interlayer cones provide Dirac mirrored we the tBLG, superlat- revealing quasicrystalline by of Moreover, tice. physics intriguing 30 the quasicrytalline of investigating with realization 30 The superlattice grown order. incommensurate successfully for have example we summary, In heterostructures Waals mecha- Conclusions der van this other Thus, to (33). tBLG. applied graphene beyond analysis be commensurate same can those nism the in cones following appear- in Dirac the systems for mode replica account structure, to of R able incommensurate ance also observed the is mechanism for the scattering only this with Not consistent spectrum. Raman fully at the is one from the mechanism scattered to identical is is cone 3D cone Fig. Dirac mirrored (see the for tour 30 xeietlosrain.Frto l,temmnu fthe of momentum valley the point all, Dirac of mirrored First observations. experimental hscrepnst h ipetcs nEq. in case simplest the to corresponds G This 6C. at Fig. valley momentum opposite elec- layer for other mechanism Applying scattering respectively. layer a counterpart, 30 propose in its trons we and condition, layer this same the of and layer, one of layer one in system: states bilayer Bloch twisted two the for condition coupling general a (see pertur- included process second-order further with scattering model bation tight-binding Umklapp a up incommensurate generalized build we a (33), in on cones Based structure. Dirac mirrored commensu- 30 of the Moir emergence in long-ranged the with appearing system cones graphene Dirac rate satellite the Unlike the of stability Discussion and incommensurate physics provides the the still or into 30 perfect, cell insights not large theoretical although infinitely some calculation, an holds Our still of structure. above limit suggest- mentioned the conclusion similar, at calculation remains the structure that electronic ing the and details), 30 5 like of supercell, stability larger a 30 to grown sample– olating of positively incoherent stability contributing and sample, the weak our to as with such systems interaction zero in substrate to exist (due thus critical could less is substrate Pt the at where u ◦ ◦ ◦ = tL niae nuuulsatrn ehns nthis in mechanism scattering unusual an indicates -tBLG -tBLG. layer 0 K Γ K k and 0 0 1 u 0 0 ◦ on,adpaemthn ewe rpeeadthe and graphene between matching phase and point, ◦ and G ◦ ssatrdb n eirclltievco fthe of vector lattice reciprocal one by scattered is G yjs n eirclltievco fteother the of vector lattice reciprocal one just by tL:Drccn tthe at cone Dirac -tBLG: G 30 d 30 ◦ ,sgetn httemroe Dirac mirrored the that suggesting 4C), Fig. or ◦ h rpeesml a bandb neln one annealing by obtained was sample graphene The tL sfrhrehne (see enhanced further is -tBLG k = eod h smer fteitniycon- intensity the of asymmetry the Second, . ◦ 2 u G omn irrdDrccone Dirac mirrored a forming , G u r aevcoso h w lc ttsin states Bloch two the of vectors wave are 30 and ◦ uhamcaimi ofimdb three by confirmed is mechanism a Such . k K 1 u R = G scnetdt h te graphene other the to connected is d ◦ k ◦ tL nteP usrt.Extrap- substrate. Pt the on -tBLG K ◦ 2 u IAppendix SI .A hshg eprtr,tecarbon the temperature, high this At C. r h eirclltievectors lattice reciprocal the are tL rvdsopruiisfor opportunities provides -tBLG − 0 K on.Tid uhascattering a such Third, point. G 0 u ◦ × + sshmtclysonin shown schematically as ,7 5, K G 0 d NSLts Articles Latest PNAS k on foegraphene one of point o eal)adreveal and details) for vco) hrfr,it Therefore, -vector). × ,ad16 and 7, ´ atr (30–32), pattern e ◦ tL,atypical a -tBLG, IAppendix SI K R × erthe near 6 the 16, | 2 f6 of 5 [2] for for K ◦ - 0

APPLIED PHYSICAL SCIENCES impurities would segregate from the bulk to the surface, forming the compression from the lower layer. Extensive convergence tests have graphene film. been performed in terms of vacuum, film thickness, and k-point sampling. Although our calculations were based on one periodic structure, such effects ARPES and NanoARPES. Conventional ARPES measurements were performed would be extrapolated to nonperiodic structures of 30◦-tBLG. at the home laboratory with a Helium discharge lamp and beamline 10.0.1 of Advanced Light Source (ALS) with synchrotron radiation source. The ACKNOWLEDGMENTS. This work is supported by Ministry of Science and NanoARPES measurements were performed at the analysis nano-spot angle- Technology of China Grants 2016YFA0303004 and 2015CB921001, National resolved photoemission spectroscopy (ANTARES) endstation of Synchrotron Natural Science Foundation of China Grants 11334006 and 11725418, Sci- SOLEIL, in France, with a lateral spatial resolution of 120 nm. The sample ence Challenge Project 20164500122, and Beijing Advanced Innovation temperature was kept at 20 K for measurements at ALS and 40 K for those Center for Future Chip (ICFC). The theoretical work is supported by the at SOLEIL. General Research Fund (Grant 2130490) from Research Grants Council in Hong Kong. The Synchrotron SOLEIL is supported by the Center National 0 Calculations. The calculations were based on VASP code (34, 35) with de la Recherche Scientifique (CNRS) and the Commissariat a` l Energie Atom- ique et aux Energies Alternatives (CEA), France. This work is supported by plane wave basis set (36, 37). We adopted opt86b-vdW functional (38, a public grant overseen by the French National Research Agency (ANR) as 39)√ to include√ the van der Waals interactions. A periodic (5 × 5) on the 0 ◦ ◦ part of the “Investissements d Avenir” program (Labex NanoSaclay, refer- (3 3×3 3)R30 structure was created to simulate the 30 -tBLG because ence ANR-10-LABX-0035) as well as the French Ministre des affaires trangres it is impossible to directly simulate the very large nonperiodic bilayer. The et europennes (MAEE), the Center National de la Recherche Scientifique lower layer is slightly compressed about 1.8% to fit the Pt lattice, and (CNRS) through Information and Communication Technology (ICT)-ASIA the upper layer is slightly stretched about 2% to partially compensate for Program Grant 3226/DGM/ATT/RECH.

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